def __init__(self, ref_el, degree, shape=tuple()):
if shape == tuple():
num_components = 1
else:
flat_shape = numpy.ravel(shape)
num_components = numpy.prod(flat_shape)
num_exp_functions = expansions.polynomial_dimension(ref_el, degree)
num_members = num_components * num_exp_functions
embedded_degree = degree
expansion_set = expansions.get_expansion_set(ref_el)
sd = ref_el.get_spatial_dimension()
# set up coefficients
coeffs_shape = tuple([num_members] + list(shape) + [num_exp_functions])
coeffs = numpy.zeros(coeffs_shape, "d")
# use functional's index_iterator function
cur_bf = 0
if shape == tuple():
coeffs = numpy.eye(num_members)
else:
for idx in index_iterator(shape):
n = expansions.polynomial_dimension(ref_el, embedded_degree)
for exp_bf in range(n):
cur_idx = tuple([cur_bf] + list(idx) + [exp_bf])
coeffs[cur_idx] = 1.0
cur_bf += 1
# construct dmats
if degree == 0:
dmats = [numpy.array([[0.0]], "d") for i in range(sd)]
else:
pts = ref_el.make_points(sd, 0, degree + sd + 1)
v = numpy.transpose(expansion_set.tabulate(degree, pts))
vinv = numpy.linalg.inv(v)
dv = expansion_set.tabulate_derivatives(degree, pts)
dtildes = [[[a[1][i] for a in dvrow] for dvrow in dv]
for i in range(sd)]
dmats = [
numpy.dot(vinv, numpy.transpose(dtilde)) for dtilde in dtildes
]
PolynomialSet.__init__(self, ref_el, degree, embedded_degree,
expansion_set, coeffs, dmats)
def __init__(self, ref_el, degree, shape=tuple()):
if shape == tuple():
num_components = 1
else:
flat_shape = numpy.ravel(shape)
num_components = numpy.prod(flat_shape)
num_exp_functions = expansions.polynomial_dimension(ref_el, degree)
num_members = num_components * num_exp_functions
embedded_degree = degree
expansion_set = expansions.get_expansion_set(ref_el)
sd = ref_el.get_spatial_dimension()
# set up coefficients
coeffs_shape = tuple([num_members] + list(shape) + [num_exp_functions])
coeffs = numpy.zeros(coeffs_shape, "d")
# use functional's index_iterator function
cur_bf = 0
if shape == tuple():
coeffs = numpy.eye(num_members)
else:
for idx in index_iterator(shape):
n = expansions.polynomial_dimension(ref_el, embedded_degree)
for exp_bf in range(n):
cur_idx = tuple([cur_bf] + list(idx) + [exp_bf])
coeffs[cur_idx] = 1.0
cur_bf += 1
# construct dmats
if degree == 0:
dmats = [numpy.array([[0.0]], "d") for i in range(sd)]
else:
pts = ref_el.make_points(sd, 0, degree + sd + 1)
v = numpy.transpose(expansion_set.tabulate(degree, pts))
vinv = numpy.linalg.inv(v)
dv = expansion_set.tabulate_derivatives(degree, pts)
dtildes = [[[a[1][i] for a in dvrow] for dvrow in dv]
for i in range(sd)]
dmats = [numpy.dot(vinv, numpy.transpose(dtilde))
for dtilde in dtildes]
PolynomialSet.__init__(self, ref_el, degree, embedded_degree,
expansion_set, coeffs, dmats)
def __init__(self, ref_el, degree):
entity_ids = {}
nodes = []
sd = ref_el.get_spatial_dimension()
t = ref_el.get_topology()
# codimension 1 facets
for i in range(len(t[sd - 1])):
pts_cur = ref_el.make_points(sd - 1, i, sd + degree)
for j in range(len(pts_cur)):
pt_cur = pts_cur[j]
f = functional.PointScaledNormalEvaluation(ref_el, i, pt_cur)
nodes.append(f)
# internal nodes. Let's just use points at a lattice
if degree > 0:
cpe = functional.ComponentPointEvaluation
pts = ref_el.make_points(sd, 0, degree + sd)
for d in range(sd):
for i in range(len(pts)):
l_cur = cpe(ref_el, d, (sd, ), pts[i])
nodes.append(l_cur)
# Q = quadrature.make_quadrature(ref_el, 2 * ( degree + 1 ))
# qpts = Q.get_points()
# Pkm1 = polynomial_set.ONPolynomialSet(ref_el, degree - 1)
# zero_index = tuple([0 for i in range(sd)])
# Pkm1_at_qpts = Pkm1.tabulate(qpts)[zero_index]
# for d in range(sd):
# for i in range(Pkm1_at_qpts.shape[0]):
# phi_cur = Pkm1_at_qpts[i, :]
# l_cur = functional.IntegralMoment(ref_el, Q, phi_cur, (d,), (sd,))
# nodes.append(l_cur)
# sets vertices (and in 3d, edges) to have no nodes
for i in range(sd - 1):
entity_ids[i] = {}
for j in range(len(t[i])):
entity_ids[i][j] = []
cur = 0
# set codimension 1 (edges 2d, faces 3d) dof
pts_facet_0 = ref_el.make_points(sd - 1, 0, sd + degree)
pts_per_facet = len(pts_facet_0)
entity_ids[sd - 1] = {}
for i in range(len(t[sd - 1])):
entity_ids[sd - 1][i] = list(range(cur, cur + pts_per_facet))
cur += pts_per_facet
# internal nodes, if applicable
entity_ids[sd] = {0: []}
if degree > 0:
num_internal_nodes = expansions.polynomial_dimension(
ref_el, degree - 1)
entity_ids[sd][0] = list(range(cur, cur + num_internal_nodes * sd))
super(RTDualSet, self).__init__(nodes, ref_el, entity_ids)
def __init__(self, ref_el, degree):
entity_ids = {}
nodes = []
sd = ref_el.get_spatial_dimension()
t = ref_el.get_topology()
# codimension 1 facets
for i in range(len(t[sd - 1])):
pts_cur = ref_el.make_points(sd - 1, i, sd + degree)
for j in range(len(pts_cur)):
pt_cur = pts_cur[j]
f = functional.PointScaledNormalEvaluation(ref_el, i, pt_cur)
nodes.append(f)
# internal nodes. Let's just use points at a lattice
if degree > 0:
cpe = functional.ComponentPointEvaluation
pts = ref_el.make_points(sd, 0, degree + sd)
for d in range(sd):
for i in range(len(pts)):
l_cur = cpe(ref_el, d, (sd,), pts[i])
nodes.append(l_cur)
# Q = quadrature.make_quadrature(ref_el, 2 * ( degree + 1 ))
# qpts = Q.get_points()
# Pkm1 = polynomial_set.ONPolynomialSet(ref_el, degree - 1)
# zero_index = tuple([0 for i in range(sd)])
# Pkm1_at_qpts = Pkm1.tabulate(qpts)[zero_index]
# for d in range(sd):
# for i in range(Pkm1_at_qpts.shape[0]):
# phi_cur = Pkm1_at_qpts[i, :]
# l_cur = functional.IntegralMoment(ref_el, Q, phi_cur, (d,), (sd,))
# nodes.append(l_cur)
# sets vertices (and in 3d, edges) to have no nodes
for i in range(sd - 1):
entity_ids[i] = {}
for j in range(len(t[i])):
entity_ids[i][j] = []
cur = 0
# set codimension 1 (edges 2d, faces 3d) dof
pts_facet_0 = ref_el.make_points(sd - 1, 0, sd + degree)
pts_per_facet = len(pts_facet_0)
entity_ids[sd - 1] = {}
for i in range(len(t[sd - 1])):
entity_ids[sd - 1][i] = list(range(cur, cur + pts_per_facet))
cur += pts_per_facet
# internal nodes, if applicable
entity_ids[sd] = {0: []}
if degree > 0:
num_internal_nodes = expansions.polynomial_dimension(ref_el,
degree - 1)
entity_ids[sd][0] = list(range(cur, cur + num_internal_nodes * sd))
super(RTDualSet, self).__init__(nodes, ref_el, entity_ids)
def RTSpace(ref_el, deg):
"""Constructs a basis for the the Raviart-Thomas space
(P_k)^d + P_k x"""
sd = ref_el.get_spatial_dimension()
vec_Pkp1 = polynomial_set.ONPolynomialSet(ref_el, deg + 1, (sd,))
dimPkp1 = expansions.polynomial_dimension(ref_el, deg + 1)
dimPk = expansions.polynomial_dimension(ref_el, deg)
dimPkm1 = expansions.polynomial_dimension(ref_el, deg - 1)
vec_Pk_indices = list(chain(*(range(i * dimPkp1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pk_from_Pkp1 = vec_Pkp1.take(vec_Pk_indices)
Pkp1 = polynomial_set.ONPolynomialSet(ref_el, deg + 1)
PkH = Pkp1.take(list(range(dimPkm1, dimPk)))
Q = quadrature.make_quadrature(ref_el, 2 * deg + 2)
# have to work on this through "tabulate" interface
# first, tabulate PkH at quadrature points
Qpts = numpy.array(Q.get_points())
Qwts = numpy.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkH_at_Qpts = PkH.tabulate(Qpts)[zero_index]
Pkp1_at_Qpts = Pkp1.tabulate(Qpts)[zero_index]
PkHx_coeffs = numpy.zeros((PkH.get_num_members(),
sd,
Pkp1.get_num_members()), "d")
for i in range(PkH.get_num_members()):
for j in range(sd):
fooij = PkH_at_Qpts[i, :] * Qpts[:, j] * Qwts
PkHx_coeffs[i, j, :] = numpy.dot(Pkp1_at_Qpts, fooij)
PkHx = polynomial_set.PolynomialSet(ref_el,
deg,
deg + 1,
vec_Pkp1.get_expansion_set(),
PkHx_coeffs,
vec_Pkp1.get_dmats())
return polynomial_set.polynomial_set_union_normalized(vec_Pk_from_Pkp1, PkHx)
def RTSpace(ref_el, deg):
"""Constructs a basis for the the Raviart-Thomas space
(P_k)^d + P_k x"""
sd = ref_el.get_spatial_dimension()
vec_Pkp1 = polynomial_set.ONPolynomialSet(ref_el, deg + 1, (sd,))
dimPkp1 = expansions.polynomial_dimension(ref_el, deg + 1)
dimPk = expansions.polynomial_dimension(ref_el, deg)
dimPkm1 = expansions.polynomial_dimension(ref_el, deg - 1)
vec_Pk_indices = list(chain(*(range(i * dimPkp1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pk_from_Pkp1 = vec_Pkp1.take(vec_Pk_indices)
Pkp1 = polynomial_set.ONPolynomialSet(ref_el, deg + 1)
PkH = Pkp1.take(list(range(dimPkm1, dimPk)))
Q = quadrature.make_quadrature(ref_el, 2 * deg + 2)
# have to work on this through "tabulate" interface
# first, tabulate PkH at quadrature points
Qpts = numpy.array(Q.get_points())
Qwts = numpy.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkH_at_Qpts = PkH.tabulate(Qpts)[zero_index]
Pkp1_at_Qpts = Pkp1.tabulate(Qpts)[zero_index]
PkHx_coeffs = numpy.zeros((PkH.get_num_members(),
sd,
Pkp1.get_num_members()), "d")
for i in range(PkH.get_num_members()):
for j in range(sd):
fooij = PkH_at_Qpts[i, :] * Qpts[:, j] * Qwts
PkHx_coeffs[i, j, :] = numpy.dot(Pkp1_at_Qpts, fooij)
PkHx = polynomial_set.PolynomialSet(ref_el,
deg,
deg + 1,
vec_Pkp1.get_expansion_set(),
PkHx_coeffs,
vec_Pkp1.get_dmats())
return polynomial_set.polynomial_set_union_normalized(vec_Pk_from_Pkp1, PkHx)
def __init__(self, ref_el, degree, size=None):
sd = ref_el.get_spatial_dimension()
if size is None:
size = sd
shape = (size, size)
num_exp_functions = expansions.polynomial_dimension(ref_el, degree)
num_components = size * (size + 1) // 2
num_members = num_components * num_exp_functions
embedded_degree = degree
expansion_set = expansions.get_expansion_set(ref_el)
# set up coefficients for symmetric tensors
coeffs_shape = tuple([num_members] + list(shape) + [num_exp_functions])
coeffs = numpy.zeros(coeffs_shape, "d")
cur_bf = 0
for [i, j] in index_iterator(shape):
n = expansions.polynomial_dimension(ref_el, embedded_degree)
if i == j:
for exp_bf in range(n):
cur_idx = tuple([cur_bf] + [i, j] + [exp_bf])
coeffs[cur_idx] = 1.0
cur_bf += 1
elif i < j:
for exp_bf in range(n):
cur_idx = tuple([cur_bf] + [i, j] + [exp_bf])
coeffs[cur_idx] = 1.0
cur_idx = tuple([cur_bf] + [j, i] + [exp_bf])
coeffs[cur_idx] = 1.0
cur_bf += 1
# construct dmats. this is the same as ONPolynomialSet.
pts = ref_el.make_points(sd, 0, degree + sd + 1)
v = numpy.transpose(expansion_set.tabulate(degree, pts))
vinv = numpy.linalg.inv(v)
dv = expansion_set.tabulate_derivatives(degree, pts)
dtildes = [[[a[1][i] for a in dvrow] for dvrow in dv]
for i in range(sd)]
dmats = [
numpy.dot(vinv, numpy.transpose(dtilde)) for dtilde in dtildes
]
PolynomialSet.__init__(self, ref_el, degree, embedded_degree,
expansion_set, coeffs, dmats)
def __init__(self, ref_el, degree, size=None):
sd = ref_el.get_spatial_dimension()
if size is None:
size = sd
shape = (size, size)
num_exp_functions = expansions.polynomial_dimension(ref_el, degree)
num_components = size * (size + 1) // 2
num_members = num_components * num_exp_functions
embedded_degree = degree
expansion_set = expansions.get_expansion_set(ref_el)
# set up coefficients for symmetric tensors
coeffs_shape = tuple([num_members] + list(shape) + [num_exp_functions])
coeffs = numpy.zeros(coeffs_shape, "d")
cur_bf = 0
for [i, j] in index_iterator(shape):
n = expansions.polynomial_dimension(ref_el, embedded_degree)
if i == j:
for exp_bf in range(n):
cur_idx = tuple([cur_bf] + [i, j] + [exp_bf])
coeffs[cur_idx] = 1.0
cur_bf += 1
elif i < j:
for exp_bf in range(n):
cur_idx = tuple([cur_bf] + [i, j] + [exp_bf])
coeffs[cur_idx] = 1.0
cur_idx = tuple([cur_bf] + [j, i] + [exp_bf])
coeffs[cur_idx] = 1.0
cur_bf += 1
# construct dmats. this is the same as ONPolynomialSet.
pts = ref_el.make_points(sd, 0, degree + sd + 1)
v = numpy.transpose(expansion_set.tabulate(degree, pts))
vinv = numpy.linalg.inv(v)
dv = expansion_set.tabulate_derivatives(degree, pts)
dtildes = [[[a[1][i] for a in dvrow] for dvrow in dv]
for i in range(sd)]
dmats = [numpy.dot(vinv, numpy.transpose(dtilde)) for dtilde in dtildes]
PolynomialSet.__init__(self, ref_el, degree, embedded_degree,
expansion_set, coeffs, dmats)
def __init__(self, ref_el, degree, variant, quad_deg):
entity_ids = {}
nodes = []
sd = ref_el.get_spatial_dimension()
t = ref_el.get_topology()
if variant == "integral":
facet = ref_el.get_facet_element()
# Facet nodes are \int_F v\cdot n p ds where p \in P_{q-1}
# degree is q - 1
Q = quadrature.make_quadrature(facet, quad_deg)
Pq = polynomial_set.ONPolynomialSet(facet, degree)
Pq_at_qpts = Pq.tabulate(Q.get_points())[tuple([0] * (sd - 1))]
for f in range(len(t[sd - 1])):
for i in range(Pq_at_qpts.shape[0]):
phi = Pq_at_qpts[i, :]
nodes.append(
functional.IntegralMomentOfScaledNormalEvaluation(
ref_el, Q, phi, f))
# internal nodes. These are \int_T v \cdot p dx where p \in P_{q-2}^d
if degree > 0:
Q = quadrature.make_quadrature(ref_el, quad_deg)
qpts = Q.get_points()
Pkm1 = polynomial_set.ONPolynomialSet(ref_el, degree - 1)
zero_index = tuple([0 for i in range(sd)])
Pkm1_at_qpts = Pkm1.tabulate(qpts)[zero_index]
for d in range(sd):
for i in range(Pkm1_at_qpts.shape[0]):
phi_cur = Pkm1_at_qpts[i, :]
l_cur = functional.IntegralMoment(
ref_el, Q, phi_cur, (d, ), (sd, ))
nodes.append(l_cur)
elif variant == "point":
# codimension 1 facets
for i in range(len(t[sd - 1])):
pts_cur = ref_el.make_points(sd - 1, i, sd + degree)
for j in range(len(pts_cur)):
pt_cur = pts_cur[j]
f = functional.PointScaledNormalEvaluation(
ref_el, i, pt_cur)
nodes.append(f)
# internal nodes. Let's just use points at a lattice
if degree > 0:
cpe = functional.ComponentPointEvaluation
pts = ref_el.make_points(sd, 0, degree + sd)
for d in range(sd):
for i in range(len(pts)):
l_cur = cpe(ref_el, d, (sd, ), pts[i])
nodes.append(l_cur)
# sets vertices (and in 3d, edges) to have no nodes
for i in range(sd - 1):
entity_ids[i] = {}
for j in range(len(t[i])):
entity_ids[i][j] = []
cur = 0
# set codimension 1 (edges 2d, faces 3d) dof
pts_facet_0 = ref_el.make_points(sd - 1, 0, sd + degree)
pts_per_facet = len(pts_facet_0)
entity_ids[sd - 1] = {}
for i in range(len(t[sd - 1])):
entity_ids[sd - 1][i] = list(range(cur, cur + pts_per_facet))
cur += pts_per_facet
# internal nodes, if applicable
entity_ids[sd] = {0: []}
if degree > 0:
num_internal_nodes = expansions.polynomial_dimension(
ref_el, degree - 1)
entity_ids[sd][0] = list(range(cur, cur + num_internal_nodes * sd))
super(RTDualSet, self).__init__(nodes, ref_el, entity_ids)
def NedelecSpace3D(ref_el, k):
"""Constructs a nodal basis for the 3d first-kind Nedelec space"""
sd = ref_el.get_spatial_dimension()
if sd != 3:
raise Exception("NedelecSpace3D requires 3d reference element")
vec_Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1, (sd, ))
dimPkp1 = expansions.polynomial_dimension(ref_el, k + 1)
dimPk = expansions.polynomial_dimension(ref_el, k)
if k > 0:
dimPkm1 = expansions.polynomial_dimension(ref_el, k - 1)
else:
dimPkm1 = 0
vec_Pk_indices = list(
chain(*(range(i * dimPkp1, i * dimPkp1 + dimPk) for i in range(sd))))
vec_Pk = vec_Pkp1.take(vec_Pk_indices)
vec_Pke_indices = list(
chain(*(range(i * dimPkp1 + dimPkm1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pke = vec_Pkp1.take(vec_Pke_indices)
Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1)
Q = quadrature.make_quadrature(ref_el, 2 * (k + 1))
Qpts = numpy.array(Q.get_points())
Qwts = numpy.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkCrossXcoeffs = numpy.zeros(
(vec_Pke.get_num_members(), sd, Pkp1.get_num_members()), "d")
Pke_qpts = vec_Pke.tabulate(Qpts)[zero_index]
Pkp1_at_Qpts = Pkp1.tabulate(Qpts)[zero_index]
for i in range(vec_Pke.get_num_members()):
for j in range(sd): # vector components
qwts_cur_bf_val = (
Qpts[:, (j + 2) % 3] * Pke_qpts[i, (j + 1) % 3, :] -
Qpts[:, (j + 1) % 3] * Pke_qpts[i, (j + 2) % 3, :]) * Qwts
PkCrossXcoeffs[i, j, :] = numpy.dot(Pkp1_at_Qpts, qwts_cur_bf_val)
# for k in range( Pkp1.get_num_members() ):
# PkCrossXcoeffs[i,j,k] = sum( Qwts * cur_bf_val * Pkp1_at_Qpts[k,:] )
# for l in range( len( Qpts ) ):
# cur_bf_val = Qpts[l][(j+2)%3] \
# * Pke_qpts[i,(j+1)%3,l] \
# - Qpts[l][(j+1)%3] \
# * Pke_qpts[i,(j+2)%3,l]
# PkCrossXcoeffs[i,j,k] += Qwts[l] \
# * cur_bf_val \
# * Pkp1_at_Qpts[k,l]
PkCrossX = polynomial_set.PolynomialSet(ref_el, k + 1, k + 1,
vec_Pkp1.get_expansion_set(),
PkCrossXcoeffs,
vec_Pkp1.get_dmats())
return polynomial_set.polynomial_set_union_normalized(vec_Pk, PkCrossX)
def NedelecSpace2D(ref_el, k):
"""Constructs a basis for the 2d H(curl) space of the first kind
which is (P_k)^2 + P_k rot( x )"""
sd = ref_el.get_spatial_dimension()
if sd != 2:
raise Exception("NedelecSpace2D requires 2d reference element")
vec_Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1, (sd, ))
dimPkp1 = expansions.polynomial_dimension(ref_el, k + 1)
dimPk = expansions.polynomial_dimension(ref_el, k)
dimPkm1 = expansions.polynomial_dimension(ref_el, k - 1)
vec_Pk_indices = list(
chain(*(range(i * dimPkp1, i * dimPkp1 + dimPk) for i in range(sd))))
vec_Pk_from_Pkp1 = vec_Pkp1.take(vec_Pk_indices)
Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1)
PkH = Pkp1.take(list(range(dimPkm1, dimPk)))
Q = quadrature.make_quadrature(ref_el, 2 * k + 2)
Qpts = numpy.array(Q.get_points())
Qwts = numpy.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkH_at_Qpts = PkH.tabulate(Qpts)[zero_index]
Pkp1_at_Qpts = Pkp1.tabulate(Qpts)[zero_index]
PkH_crossx_coeffs = numpy.zeros(
(PkH.get_num_members(), sd, Pkp1.get_num_members()), "d")
def rot_x_foo(a):
if a == 0:
return 1, 1.0
elif a == 1:
return 0, -1.0
for i in range(PkH.get_num_members()):
for j in range(sd):
(ind, sign) = rot_x_foo(j)
for k in range(Pkp1.get_num_members()):
PkH_crossx_coeffs[i, j, k] = sign * sum(
Qwts * PkH_at_Qpts[i, :] * Qpts[:, ind] *
Pkp1_at_Qpts[k, :])
# for l in range( len( Qpts ) ):
# PkH_crossx_coeffs[i,j,k] += Qwts[ l ] \
# * PkH_at_Qpts[i,l] \
# * Qpts[l][ind] \
# * Pkp1_at_Qpts[k,l] \
# * sign
PkHcrossx = polynomial_set.PolynomialSet(ref_el, k + 1, k + 1,
vec_Pkp1.get_expansion_set(),
PkH_crossx_coeffs,
vec_Pkp1.get_dmats())
return polynomial_set.polynomial_set_union_normalized(
vec_Pk_from_Pkp1, PkHcrossx)
def NedelecSpace3D(ref_el, k):
"""Constructs a nodal basis for the 3d first-kind Nedelec space"""
sd = ref_el.get_spatial_dimension()
if sd != 3:
raise Exception("NedelecSpace3D requires 3d reference element")
vec_Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1, (sd,))
dimPkp1 = expansions.polynomial_dimension(ref_el, k + 1)
dimPk = expansions.polynomial_dimension(ref_el, k)
if k > 0:
dimPkm1 = expansions.polynomial_dimension(ref_el, k - 1)
else:
dimPkm1 = 0
vec_Pk_indices = list(chain(*(range(i * dimPkp1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pk = vec_Pkp1.take(vec_Pk_indices)
vec_Pke_indices = list(chain(*(range(i * dimPkp1 + dimPkm1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pke = vec_Pkp1.take(vec_Pke_indices)
Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1)
Q = quadrature.make_quadrature(ref_el, 2 * (k + 1))
Qpts = numpy.array(Q.get_points())
Qwts = numpy.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkCrossXcoeffs = numpy.zeros((vec_Pke.get_num_members(),
sd,
Pkp1.get_num_members()), "d")
Pke_qpts = vec_Pke.tabulate(Qpts)[zero_index]
Pkp1_at_Qpts = Pkp1.tabulate(Qpts)[zero_index]
for i in range(vec_Pke.get_num_members()):
for j in range(sd): # vector components
qwts_cur_bf_val = (
Qpts[:, (j + 2) % 3] * Pke_qpts[i, (j + 1) % 3, :] -
Qpts[:, (j + 1) % 3] * Pke_qpts[i, (j + 2) % 3, :]) * Qwts
PkCrossXcoeffs[i, j, :] = numpy.dot(Pkp1_at_Qpts, qwts_cur_bf_val)
# for k in range( Pkp1.get_num_members() ):
# PkCrossXcoeffs[i,j,k] = sum( Qwts * cur_bf_val * Pkp1_at_Qpts[k,:] )
# for l in range( len( Qpts ) ):
# cur_bf_val = Qpts[l][(j+2)%3] \
# * Pke_qpts[i,(j+1)%3,l] \
# - Qpts[l][(j+1)%3] \
# * Pke_qpts[i,(j+2)%3,l]
# PkCrossXcoeffs[i,j,k] += Qwts[l] \
# * cur_bf_val \
# * Pkp1_at_Qpts[k,l]
PkCrossX = polynomial_set.PolynomialSet(ref_el,
k + 1,
k + 1,
vec_Pkp1.get_expansion_set(),
PkCrossXcoeffs,
vec_Pkp1.get_dmats())
return polynomial_set.polynomial_set_union_normalized(vec_Pk, PkCrossX)
def NedelecSpace2D(ref_el, k):
"""Constructs a basis for the 2d H(curl) space of the first kind
which is (P_k)^2 + P_k rot( x )"""
sd = ref_el.get_spatial_dimension()
if sd != 2:
raise Exception("NedelecSpace2D requires 2d reference element")
vec_Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1, (sd,))
dimPkp1 = expansions.polynomial_dimension(ref_el, k + 1)
dimPk = expansions.polynomial_dimension(ref_el, k)
dimPkm1 = expansions.polynomial_dimension(ref_el, k - 1)
vec_Pk_indices = list(chain(*(range(i * dimPkp1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pk_from_Pkp1 = vec_Pkp1.take(vec_Pk_indices)
Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1)
PkH = Pkp1.take(list(range(dimPkm1, dimPk)))
Q = quadrature.make_quadrature(ref_el, 2 * k + 2)
Qpts = numpy.array(Q.get_points())
Qwts = numpy.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkH_at_Qpts = PkH.tabulate(Qpts)[zero_index]
Pkp1_at_Qpts = Pkp1.tabulate(Qpts)[zero_index]
PkH_crossx_coeffs = numpy.zeros((PkH.get_num_members(),
sd,
Pkp1.get_num_members()), "d")
def rot_x_foo(a):
if a == 0:
return 1, 1.0
elif a == 1:
return 0, -1.0
for i in range(PkH.get_num_members()):
for j in range(sd):
(ind, sign) = rot_x_foo(j)
for k in range(Pkp1.get_num_members()):
PkH_crossx_coeffs[i, j, k] = sign * sum(Qwts * PkH_at_Qpts[i, :] * Qpts[:, ind] * Pkp1_at_Qpts[k, :])
# for l in range( len( Qpts ) ):
# PkH_crossx_coeffs[i,j,k] += Qwts[ l ] \
# * PkH_at_Qpts[i,l] \
# * Qpts[l][ind] \
# * Pkp1_at_Qpts[k,l] \
# * sign
PkHcrossx = polynomial_set.PolynomialSet(ref_el,
k + 1,
k + 1,
vec_Pkp1.get_expansion_set(),
PkH_crossx_coeffs,
vec_Pkp1.get_dmats())
return polynomial_set.polynomial_set_union_normalized(vec_Pk_from_Pkp1,
PkHcrossx)
